Book Review: Rational curves on algebraic varieties

Geometry of Rational Curves on Algebraic Varieties

Geometry of Rational Curves on Algebraic Varieties

Rational Curves on Varieties

Counting Rational Points on Algebraic Varieties

In these lectures we will be interested in solutions to Diophantine equations F (x1, . . . , xn) = 0, where F is an absolutely irreducible polynomial with integer coefficients, and the solutions are to satisfy (x1, . . . , xn) ∈ Z. Such an equation represents a hypersurface in A, and we may prefer to talk of integer points on this hypersurface, rather than solutions to the corresponding Diophan...

Counting Rational Points on Algebraic Varieties

For any N ≥ 2, let Z ⊂ P be a geometrically integral algebraic variety of degree d. This paper is concerned with the number NZ(B) of Q-rational points on Z which have height at most B. For any ε > 0 we establish the estimate NZ(B) = Od,ε,N (B ), provided that d ≥ 6. As indicated, the implied constant depends at most upon d, ε and N . Mathematics Subject Classification (2000): 11G35 (14G05)

Rational Curves on Minuscule Schubert Varieties

Let us denote by C the variety of lines in P3 meeting a fixed line, it is a grassmannian (and hence minuscule) Schubert variety. In [P2] we described the irreducible components of the scheme of morphisms from P1 to C and the general morphism of these irreducible components. In this text we study the scheme of morphisms from P1 to any minuscule Schubert variety X. Let us recall that we studied i...